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Kaon Physics Kaon mixing and eigenstates Neutral kaons and CP violation Neutral kaons and flavor oscillations Kaon physics . or an introduction to flavor oscillations and CP violation Harri Waltari University of Helsinki & Helsinki Institute of Physics University of Southampton & Rutherford Appleton Laboratory Autumn 2018 H. Waltari Kaon physics Kaon mixing and eigenstates Neutral kaons and CP violation Neutral kaons and flavor oscillations Contents Kaons are the lightest strange mesons and relatively long-lived. They are from the experimental point of view the easiest platform to study many of the interesting features of flavor physics with high precision. In this lecture we shall study the flavor and CP eigenstates of kaons study how CP violation emerges in kaon physics 0 study the flavor oscillations between K 0 and K This lecture corresponds to chapters 14.4, 14.5.1, 14.5.2 and 14.5.4 of Thomson's book. H. Waltari Kaon physics Kaon mixing and eigenstates Neutral kaons and CP violation Neutral kaons and flavor oscillations 0 Weak interactions mix the neutral kaon states K 0 and K Neutral kaons are produced in two different quark compositions K 0 0 (ds) and K (ds) | notice that for historical reasons, the strangeness quantum number is so defined that S(K 0) = +1, i.e. the antiquark has positive strangeness As neutral kaons propagate, they can transform between each other 0 through box diagrams, so neither K 0 nor K is an eigenstate of the full Hamiltonian H. Waltari Kaon physics Kaon mixing and eigenstates Neutral kaons and CP violation Neutral kaons and flavor oscillations The physical eigenstates have differing lifetimes Experimentally we observe that there are two different neutral kaon states with (almost) the same mass (close to 500 MeV) but different lifetimes, one having τ ' 0:1 ns and the other with τ ' 50 ns, these are called KS and KL (for short and long) We know that in the leptonic sector CP is (at least within current experimental precision) conserved so we may expect the CP eigenstates to be (at least close to) the stationary states 0 0 Kaons are JP = 0− mesons so PjK 0i = −|K 0i and PjK i = −|K i The flavor eigenstates have opposite flavor contents so 0 0 CjK 0i = eiζ jK i and CjK i = e−iζ jK 0i, where ζ is an unobservable phase If we choose ζ = π (which is a common convention) 0 0 CjK 0i = −|K i and CjK i = −|K 0i 0 0 Hence CPjK 0i = jK i and CPjK i = jK 0i H. Waltari Kaon physics Kaon mixing and eigenstates Neutral kaons and CP violation Neutral kaons and flavor oscillations 0 The CP eigenstates are combinations of K 0 and K 0 From the CP properties of K 0 and K it is easy to construct the CP eigenstates: CP eigenstates of neutral kaons 0 jK i = p1 (jK 0i + jK i) with CP = +1 and 1 2 0 jK i = p1 (jK 0i − jK i) with CP = −1 2 2 In weak interactions CP is nearly conserved so these almost coincide with the physical eigenstates KS and KL On the other hand flavor eigenstates are linear combinations of jK1i and jK2i but this combination will have a non-trivial time evolution As we shall see, some of the decays are spesific to flavor eigenstates and some to CP eigenstates, which will give us interesting probes on the kaon system H. Waltari Kaon physics Kaon mixing and eigenstates Neutral kaons and CP violation Neutral kaons and flavor oscillations Decays to pions determine the CP properties of KS and KL 0 0 + − Experimentally we see that KS decays mainly to π π or π π , 0 0 0 + 0 − while KL decays to π π π or π π π These decay modes determine the lifetimes | for the three pion decay there is less phase space available so the decay is slower As both kaons and pions are JP = 0− mesons, the pions must be in an ` = 0 state in the decay K 0 ! π0π0; π+π− to conserve angular momentum Hence the parity of the final state is P(π0π0) = (−1)`P(π0)P(π0) = 1 · (−1) · (−1) = 1, similarly for π+π− The neutral pion is a state jπ0i = p1 (uu − dd) so its C-parity is +1 2 as C transforms the pion to itself For π+π− C is equivalent to the exchange of particles and for bosons this does not change the sign 0 0 + − Hence CP(π π ) = +1 and CP(π π ) = +1 implying that KS could be identified with K1 H. Waltari Kaon physics Kaon mixing and eigenstates Neutral kaons and CP violation Neutral kaons and flavor oscillations Decays to pions determine the CP properties of KS and KL The angular momentum argument is trickier for the three pion final state ~ We first take the angular momentum of two pions labeling it L1 and then the angular momentum of the third with respect to the ~ center-of-mass of the pair (L2) ~ ~ ~ The total angular momentum is L = L1 + L2, which must be zero due to angular momentum conservation (the spins are all zero), giving jL1j = jL2j Hence the parity is (−1)L1 · (−1)L2 · (P(π0))3 = −1 H. Waltari Kaon physics Kaon mixing and eigenstates Neutral kaons and CP violation Neutral kaons and flavor oscillations Decays to pions determine the CP properties of KS and KL The C-parities of the three-pion final states are +1 by the same arguments as for the two pion case Hence the CP for the three pion state is negative, which implies CP(KL) = −1 Thus jKLi can be identified with jK2i In 1964 Christenson, Cronin, Fitch and Turlay observed the decay + − KL ! π π , which was the first (and together with other similar kaon decays, the only for more than 30 years) indication of CP violation in weak interactions (Fitch and Cronin got the 1980 Nobel Prize for this discovery) H. Waltari Kaon physics Kaon mixing and eigenstates Neutral kaons and CP violation Neutral kaons and flavor oscillations Particle decays can be described by complex energy eigenvalues We shall first look at kaon state vectors in the limit of CP conservation Kaons are produced as flavor eigenstates, e.g. K 0 = p1 (jK i + jK i) 2 S L The time evolution can be properly described if we attach an imaginary part to the eigenvalue of the Hamiltonian, e.g. −im t−Γ t=2 jKS (t)i = jKS (0)ie S S −Γ t In such a case hKS (t)jKS (t)i = e S so the wave function dies out in the timescale τS = 1=ΓS As there is a factor of 500 between the lifetimes of KS and KL, after a reasonably long propagation time a beam initially in a flavor eigenstate has become a pure jKLi state H. Waltari Kaon physics Kaon mixing and eigenstates Neutral kaons and CP violation Neutral kaons and flavor oscillations CP is nearly conserved | only a tiny fraction of decays are CP violating KS decays BR KL decays BR + − + − KS ! π π 69:2% KL ! π π 0:20% 0 0 0 0 KS ! π π 30:7% KL ! π π 0:09% 0 0 0 −8 0 0 0 KS ! π π π < 2:6 × 10 KL ! π π π 19:5% + − 0 −7 + − 0 KS ! π π π 3:5 × 10 KL ! π π π 12:5% − + − + KS ! π e νe 0:03% KL ! π e νe 20:3% + − + − KS ! π e νe 0:03% KL ! π e νe 20:3% Here the leptonic modes have equal decay rates but due to the larger total rate of KS , the branching ratios are smaller There is a clear difference in the order of magnitude of the CP violating decays, 2 orders of magnitude when kinematics are favorable and 6 orders of magnitude when they are not H. Waltari Kaon physics Kaon mixing and eigenstates Neutral kaons and CP violation Neutral kaons and flavor oscillations CP is violated both in kaon mixing and decays There are two possible sources for CP violation: The Hamiltonian can introduce CP violation in the kaon mixing so that the physical eigenstates do not coincide with the CP ones, it could also be possible for the kaon decays to be directly affected by the CP violating phase In the first case the eigenstates are parametrized as 1 1 jKS i = (jK1i + jK2i); jKLi = (jK2i + jK1i); p1 + jj2 p1 + jj2 where is a small complex parameter 0 The second case is parametrized by = Γ(K2 ! ππ)=Γ(K2 ! πππ) Both are nonzero and small and <(0/) ' 1:65 × 10−3 H. Waltari Kaon physics Kaon mixing and eigenstates Neutral kaons and CP violation Neutral kaons and flavor oscillations The Hamiltonian gets contributions from interaction potentials and decays We shall next go through the kaon mixing and see how most of the CP violation is generated. The mixing is a second order effect in weak 2 interactions, i.e. proportional to GF . The Hamiltonian equation of motion for, say, jK 0i in its rest frame @ 0 i 0 is i @t jK (t)i = (m − 2 Γ)jK (t)i The mass term is the sum of the quark masses and the potential energy of the interactions between quarks: 0 0 0 0 X hK jHW jjihjjHW jK i m = md +ms +hK jHQCD +HQED +HW jK i+ Ej − mK j The decay rate is determined by the Fermi golden rule: P 0 2 Γ = 2π f jhf jHW jK ij ρf , where the sum goes over all possible final states jf i and ρf is the corresponding density of states H.
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